The answer is: [tex] f^{-1} = 4x^2-3,\quad\text{for } x \leq 0 [/tex]
The inverse of a function [tex] f(x) [/tex] is another function, [tex] f^{-1}(x) [/tex], with the following property:
[tex] f(f^{-1}(x)) = f^{-1}(f(x)) = x [/tex]
In other words, the inverse of a function does exactly "the opposite" of what the original function does, and so if you compute them both in sequence you return to the starting point.
Think for example to a function that doubles the input, [tex] f(x)=2x [/tex], and one that halves it: [tex] f(x)= \frac{x}{2} [/tex]. Their composition is clearly the identity function [tex] f(x)=x [/tex], since you consider "twice the half of something", or "half the double of something".
In general, to invert a function [tex] y=f(x) [/tex], you have to solve the expression for [tex] x [/tex], writing an expression like [tex] x = g(y) [/tex]. If you manage to do so, then [tex] g [/tex] is the inverse of [tex] f [/tex].
In your case, you have
[tex] f(x) = y = -\frac{1}{2}\sqrt{x+3} [/tex]
Multiply both sides by [tex] -2 [/tex] to get
[tex] -2y = \sqrt{x+3} [/tex]
Square both sides to get
[tex] 4y^2 = x+3 [/tex]
Finally, subtract 3 from both sides to get
[tex] x = 4y^2 - 3 [/tex]
Since the name of the variables doesn't really have a meaning, you can say that the inverse function is
[tex] f^{-1}(x) = 4x^2 - 3 [/tex]
As for the domain of the inverse function, remember what we said ad the beginning: if the original function goes from set A (domain) to set B (codomain), then the inverse function goes from set B (domain) to set A (codomain). This means that the inverse function is defined on an element in B if and only if that element belongs to the range of the original function, i.e. the set of the elements of the codomain [tex] b \in B [/tex] such that there exists [tex] a \in A : f(a)=b [/tex]. So, we need the range of [tex] f(x) [/tex].
We know that the range of [tex] g(x)=\sqrt{x} [/tex] is [tex] [0,\infty) [/tex]. When you transform it to [tex] g(x)=\sqrt{x+3} [/tex] you simply translate the graph horizontally, so the range doesn't change. But when you multiply the function times [tex] -\frac{1}{2} [/tex] you affect both extrema of the range, turning it into [tex] (-\infty,0] [/tex], which you can simply write as [tex] x \leq 0 [/tex]
